“Richardson's theorem”的意思、由来-开放百科全书

In mathematics, Richardson's theorem establishes a limit on the extent to which an algorithm can decide whether certain mathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is undecidable whether a particular expression E satisfies the equation E = 0, and similarly undecidable whether the functions defined by expressions E and F are everywhere equal (in fact E = F if and only if E - F = 0). It was proved in 1968 by computer scientist Daniel Richardson of the University of Bath.

Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number log 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.

For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether an expression is zero.^[1]

Statement of the theorem

Let E be a set of expressions in the variable x which contains x and, as constant expressions, all rational numbers, and is such that if A(x) and B(x) are in E, then A(x) + B(x), A(x) - B(x), A(x)B(x), and A(B(x)) are also in E. Then:

Extensions

After Hilbert's Tenth Problem was solved in 1970, B. F. Caviness observed that the use of e^x and log 2 could be removed.^[3]

P. S. Wang later noted that under the same assumptions under which the question of whether there was x with A(x) < 0 was insoluble, the question of whether there was x with A(x) = 0 was also insoluble.^[4]

Miklós Laczkovich removed also the need for π and reduced the use of composition.^[5] In particular, given an expression A(x) in the ring generated by the integers, x, sin xⁿ, and sin(x sin xⁿ), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable.

By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.

See also

References

1. ^Dan Richardson and John Fitch, 1994, "The identity problem for elementary functions and constants", Proceedings of the international symposium on Symbolic and algebraic computation, pp. 85–290.
2. ^{{cite journal |zbl=0175.27404 |title=Some Undecidable Problems Involving Elementary Functions of a Real Variable |first=Daniel |last=Richardson |journal=J. Symbolic Logic |volume=33 |issue=4 |year=1968 |pages=514–520 |jstor=2271358 }}
3. ^{{cite journal |title=On Canonical Forms and Simplification |first=B. F. |last=Caviness |journal=Journal of the ACM |volume=17 |issue=2 |year=1970 |pages=385–396 |doi=10.1145/321574.321591 }}
4. ^{{cite journal |first=P. S. |last=Wang |title=The undecidability of the existence of zeros of real elementary functions |journal=Journal of the Association for Computing Machinery |volume=21 |issue=4 |year=1974 |pages=586–589 |doi=10.1145/321850.321856 }}
5. ^{{cite journal |first=Miklós |last=Laczkovich |title=The removal of π from some undecidable problems involving elementary functions |journal=Proc. Amer. Math. Soc. |volume=131 |issue=7 |year=2003 |pages=2235–2240 |doi=10.1090/S0002-9939-02-06753-9 }}

Statement of the theorem

Extensions

See also

References

Further reading

External links